A Uniqueness Theorem for the Two-Dimensional Sigma Function
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Uniqueness Theorem for the Two-Dimensional Sigma Function A. V. Domrin Received April 21, 2019; in final form, April 21, 2019; accepted October 31, 2019
Abstract. We prove that the sigma functions of Weierstrass (g = 1) and Klein (g = 2) are the unique solutions (up to multiplication by a complex constant) of the corresponding systems of 2g linear differential heat equations in a nonholonomic frame (for a function of 3g variables) that are holomorphic in a neighborhood of at least one point where all modular variables vanish. We also show that all local holomorphic solutions of these systems can be extended analytically to entire functions of angular variables. For g = 1, we give a complete description of the envelopes of holomorphy of such solutions. Key words: hyperelliptic sigma-function, systems of linear differential heat equations in a nonholonomic frame. DOI: 10.1134/S0016266320010037
1. Introduction / R, is Every lattice Λ = {n1 ω1 + n2 ω2 ∈ C | n1 , n2 ∈ Z}, where ω1 , ω2 ∈ C \ {0} and ω2 /ω1 ∈ associated with the corresponding classical Weierstrass sigma function σ(u, Λ) = σ(u | ω1 , ω2 ) := u
ω∈Λ\{0}
u 1− ω
exp
u u2 + , ω 2ω 2
which is an entire function of u with simple zeros at all points ω ∈ Λ. Note that the lattice Λ is uniquely determined by its invariants g2 (Λ) := 60
ω∈Λ\{0}
ω −4
and g3 (Λ) := 140
ω −6 ,
g23 (Λ) − 27g32 (Λ) = 0.
ω∈Λ\{0}
Indeed, Λ is the set of poles of the unique solution with a pole at u = 0 of the differential equation f (u)2 = 4f (u)3 − g2 (Λ)f (u) − g3 (Λ) (this solution is ℘(u) = −∂u2 (log σ(u, Λ)). Therefore, σ(u, Λ) may be regarded as a function σ(u, g2 , g3 ) defined on the open set {(u, g2 , g3 ) ∈ C3 | g23 − 27g32 = 0} and related to the family of plane curves y 2 = 4x3 − g2 x − g3 . The following fact was observed by Weierstrass [1; Sec. 5], who used it in his construction of the theory of elliptic functions. The function σ(u, g2 , g3 ) extends analytically to an entire function on the whole space C3 and satisfies the following system of linear differential equations in C3 (where ∂2 := ∂/∂g2 , ∂3 := ∂/∂g3 , and ∂u := ∂/∂u): 4g2 ∂2 ϕ + 6g3 ∂3 ϕ = u∂u ϕ − ϕ, 6g3 ∂2 ϕ + 13 g22 ∂3 ϕ = 12 ∂u2 ϕ + 16 g2 u2 ϕ.
(1)
The first equation of system (1) is equivalent to the homogeneity property ϕ(u; g2 , g3 ) = tϕ(u/t; t4 g2 , series in u, g2 , and g3 t6 g3 ), t ∈ C \ {0}. Therefore, all solutions of (1) in the form of a formal power m n 4m+6n+1 divisible by u can be found by substituting the expression ϕ(u; g2 , g3 ) = ∞ m,n=0 amn g2 g3 u into the second equation of the system. The resulting recursive formula for the coefficients amn proves the following assertion. c 2020 by Pleiades Publishing, Ltd. 0016–2663/19/5303–0174
21
Theorem A [1; Sec. 5]. The Weierstrass sigma function σ(u; g2 , g3 ) is a unique solution of system (1) with initial condition ϕ(u; 0, 0) = u in the class of formal power series in the variables u, g2 , and g3 . The notion of a sigma function related to higher-genus curves was introduced by Klein [2] in 1886. Since then it has b
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